Abel-Ruffini Theorem

Is the Abel-Ruffini Theorem essentially equivalent to saying that the set of all complex numbers constructable by a concatenation of field operations and n-roots of rational numbers is a proper subset of the algebraic numbers?

But that field generated by the coefficients of the given polynomial may very well contain transcendental numbers as well. If $K$ is the field generated by the coefficients of such a polynomial Abel-Ruffini says that the zeros of that polynomial, while necessarily transcendental numbers themselves, do not even belong to a root tower extension of $K$.
• Oh, so for algebraic numbers we think of polynomials in $\mathbb{Q}[x]$, but for Abel Ruffini the polynomials can have coefficients in any field? – Robearz Jun 24 '14 at 18:41
• Correct. I guess for the purposes of quintic polynomials we may want the field to have characteristic $>5$, but the field may, for example, be an extension of $\Bbb{C}$. – Jyrki Lahtonen Jun 24 '14 at 18:49